Understanding Basic Music Theory

Chapter 3

The Physical Basis

3.1 Acoustics for Music Theory1

3.1.1 Music is Organized Sound Waves

Music is sound that's organized by people on purpose, to dance to, to tell a story, to make other

people feel a certain way, or just to sound pretty or be entertaining. Music is organized on many

di erent levels. Sounds can be arranged into melodies (Section 2.3), harmonies (Section 2.5), rhythms

(Section 2.1), textures (Section 2.4) and phrases (Section 2.3.4). Beats (Section 1.2.3.1), measures

(Section 1.1.1.1), cadences (Section 5.6), and form (Section 5.7) all help to keep the music organized

and understandable. But the most basic way that music is organized is by arranging the actual

sound waves themselves so that the sounds are interesting and pleasant and go well together.

A rhythmic, organized set of thuds and crashes is perfectly good music - think of your favorite

drum solo - but many musical instruments are designed speci cally to produce the regular, evenly

spaced sound waves that we hear as particular pitches (Section 1.1.3). Crashes, thuds, and bangs

are loud, short jumbles of lots of di erent wavelengths. These are the kinds of sound we often

call "noise", when they're random and disorganized, but as soon as they are organized in time

noise(rhythm (Section 2.1)), they begin to sound like music. (When used as a scienti c term,
continuousrefers to
sounds that are random mixtures of di erent wavelengths, not shorter crashes

and thuds.)

However, to get the melodic kind of sounds more often associated with music, the sound waves

must themselves be organized and regular, not random mixtures. Most of the sounds we hear are

brought to our ears through the air. A movement of an object causes a disturbance of the normal

motion of the air molecules near the object. Those molecules in turn disturb other nearby molecules

out of their normal patterns of random motion, so that the disturbance itself becomes a thing that

moves through the air - a sound wave. If the movement of the object is a fast, regular vibration,

tonesthen the sound waves are also very regular. We hear such regular sound waves as , sounds

with a particular pitch (Section 1.1.3). It is this kind of sound that we most often associate with

music, and that many musical instruments are designed to make.

1This content is available online at <http://cnx.org/content/m13246/1.7/>.

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96 CHAPTER 3. THE PHYSICAL BASIS

Figure 3.1: A random jumble of sound waves is heard as a noise. A regular, evenly-spaced sound

wave is heard as a tone.

Musicians have terms that they use to describe tones. (Musicians also have other meanings
for the word "tone", but this course will stick to the "a sound with pitch" meaning.) This kind
of (regular, evenly spaced) wave is useful for things other than music, however, so scientists and
engineers also have terms that describe pitched sound waves. As we talk about where music theory
comes from, it will be very useful to know both the scienti c and the musical terms and how they
are related to each other.

For example, the closer together those evenly-spaced waves are, the higher the note sounds.
Musicians talk about the pitch (Section 1.1.3) of the sound, or name speci c notes (Section 1.1.2),
or talk about tuning (Section 6.2). Scientists and engineers, on the other hand, talk about the
frequency (pg 99) and the wavelength (pg 99) of the sound. They are all essentially talking about
the same things, but talking about them in slightly di erent ways, and using the scienti c ideas of
wavelength and frequency can help clarify some of the main ideas underlying music theory.

3.1.2 Longitudinal and Transverse Waves

So what are we talking about when we speak of sound waves? Waves are disturbances; they are

changes in something - the surface of the ocean, the air, electromagnetic elds. Normally, these

changes are travelling (except for standing waves (Section 3.2)); the disturbance is moving away

from whatever created it, in a kind of domino e ect.

transverseMost kinds of waves are waves. In a transverse wave, as the wave is moving in one

direction, it is creating a disturbance in a di erent direction. The most familiar example of this is

waves on the surface of water. As the wave travels in one direction - say south - it is creating an

up-and-down (not north-and-south) motion on the water's surface. This kind of wave is fairly easy

to draw; a line going from left-to-right has up-and-down wiggles. (See Figure 3.2.)

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Transverse and Longitudinal Waves

Figure 3.2: In water waves and other transverse waves, the ups and downs are in a di erent

direction from the forward movement of the wave. The "highs and lows" of sound waves and other

longitudinal waves are arranged in the "forward" direction.

longitudinal wavesBut sound waves are not transverse. Sound waves are . If sound waves are

moving south, the disturbance that they are creating is giving the air molecules extra north-and-

south (not east-and-west, or up-and-down) motion. If the disturbance is from a regular vibration, the

result is that the molecules end up squeezed together into evenly-spaced waves. This is very di cult

to show clearly in a diagram, so most diagrams, even diagrams of sound waves, show transverse
waves.

Longitudinal waves may also be a little di cult to imagine, because there aren't any examples

that we can see in everyday life (unless you like to play with toy slinkies). A mathematical description

might be that in longitudinal waves, the waves (the disturbances) are along the same axis as the

direction of motion of the wave; transverse waves are at right angles to the direction of motion

of the wave. If this doesn't help, try imagining yourself as one of the particles that the wave is

disturbing (a water drop on the surface of the ocean, or an air molecule). As it comes from behind

you, a transverse waves lifts you up and then drops down; a longitudinal wave coming from behind

pushes you forward and pulls you back. You can view here animations of longitudinal and transverse

waves2, single particles being disturbed by a transverse wave or by a longitudinal wave 3, and particles
being disturbed by transverse and longitudinal waves 4. (There were also some nice animations of
longitudinal waves available as of this writing at Musemath 5.)

The result of these "forward and backward" waves is that the "high point" of a sound wave

is where the air molecules are bunched together, and the "low point" is where there are fewer air

molecules. In a pitched sound, these areas of bunched molecules are very evenly spaced. In fact,

Inthey are so even, that there are some very useful things we can measure and say about them.
order to clearly show you what they are, most of the diagrams in this course will show sound waves
as if they are transverse waves.

2http://cnx.org/content/m13246/latest/Waves.swf
3http://cnx.org/content/m13246/latest/Pulses.swf
4http://cnx.org/content/m13246/latest/Translong.swf
5http://www.musemath.com

98 CHAPTER 3. THE PHYSICAL BASIS

3.1.3 Wave Amplitude and Loudness

displacementBoth transverse and longitudinal waves cause a of something: air molecules, for

example, or the surface of the ocean. The amount of displacement at any particular spot changes

as the wave passes. If there is no wave, or if the spot is in the same state it would be in if there was

no wave, there is no displacement. Displacement is biggest (furthest from "normal") at the highest

and lowest points of the wave. In a sound wave, then, there is no displacement wherever the air

molecules are at a normal density. The most displacement occurs wherever the molecules are the

most crowded or least crowded.

Displacement

Figure 3.3

amplitudeThe of the wave is a measure of the displacement: how big is the change from no

displacement to the peak of a wave? Are the waves on the lake two inches high or two feet? Are

the air molecules bunched very tightly together, with very empty spaces between the waves, or are

they barely more organized than they would be in their normal course of bouncing o of each other?

decibelsScientists measure the amplitude of sound waves in . Leaves rustling in the wind are about

10 decibels; a jet engine is about 120 decibels.

dynamic level ForteMusicians call the loudness of a note its
. (pronounced "FOR-tay") is a

pianoloud dynamic level; is soft. Dynamic levels don't correspond to a measured decibel level. An

orchestra playing "fortissimo" (which basically means "even louder than forte") is going to be quite

a bit louder than a string quartet playing "fortissimo". (See Dynamics (Section 1.3.1) for more of

the terms that musicians use to talk about loudness.) Dynamics are more of a performance issue

than a music theory issue, so amplitude doesn't need much discussion here.

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Amplitude is Loudness

Figure 3.4: The size of a wave (how much it is "piled up" at the high points) is its amplitude.

For sound waves, the bigger the amplitude, the louder the sound.

3.1.4 Wavelength, Frequency, and Pitch

The aspect of evenly-spaced sound waves that really a ects music theory is the spacing between

the waves, the distance between, for example, one high point and the next high point. This is the

wavelength, and it a ects the pitch (Section 1.1.3) of the sound; the closer together the waves are,

the higher the tone sounds.

All sound waves are travelling at about the same speed - the speed of sound. So waves with

a shorter wavelength arrive (at your ear, for example) more often (frequently) than longer waves.

frequencyThis aspect of a sound - how often a peak of a wave goes by, is called by scientists and

hertzengineers. They measure it in , which is how many peaks go by per second. People can hear

sounds that range from about 20 to about 17,000 hertz.

100 CHAPTER 3. THE PHYSICAL BASIS

Wavelength, Frequency, and Pitch

Figure 3.5: Since the sounds are travelling at about the same speed, the one with the shorter

wavelength "waves" more frequently; it has a higher frequency, or pitch. In other words, it sounds
higher.

pitchThe word that musicians use for frequency is . The shorter the wavelength, the higher the

frequency, and the higher the pitch, of the sound. In other words, short waves sound high; long

waves sound low. Instead of measuring frequencies, musicians name the pitches (Section 1.1.2) that

they use most often. They might call a note "middle C" or "second line G" or "the F sharp in

the bass clef". (See Octaves and Diatonic Music (Section 4.1) and Tuning Systems (Section 6.2) for

more on naming speci c frequencies.) These notes have frequencies (Have you heard of the "A 440"

that is used as a tuning note?), but the actual frequency of a middle C can vary a little from one

orchestra, piano, or performance, to another, so musicians usually nd it more useful to talk about

note names.

Most musicians cannot name the frequencies of any notes other than the tuning A (440 hertz).

The human ear can easily distinguish two pitches that are only one hertz apart when it hears them

both, but it is the very rare musician who can hear speci cally that a note is 442 hertz rather

than 440. So why should we bother talking about frequency, when musicians usually don't? As we

will see, the physics of sound waves - and especially frequency - a ects the most basic aspects of

music, including pitch (Section 1.1.3), tuning (Section 6.2), consonance and dissonance (Section 5.3),

harmony (Section 2.5), and timbre (Section 2.2).

3.2 Standing Waves and Musical Instruments6

3.2.1 What is a Standing Wave?

Musical tones (pg 101) are produced by musical instruments, or by the voice, which, from a physics

perspective, is a very complex wind7 instrument. So the physics of music is the physics of the kinds

of sounds these instruments can make. What kinds of sounds are these? They are tones caused by

6This content is available online at <http://cnx.org/content/m12413/1.8/>.
7"Wind Instruments: Some Basics" <http://cnx.org/content/m12364/latest/>

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standing waves produced in or on the instrument. So the properties of these standing waves, which

are always produced in very speci c groups, or series, have far-reaching e ects on music theory.

Most sound waves, including the musical sounds that actually reach our ears, are not standing

waves. Normally, when something makes a wave, the wave travels outward, gradually spreading out

and losing strength, like the waves moving away from a pebble dropped into a pond.

But when the wave encounters something, it can bounce (re ection) or be bent (refraction). In

fact, you can "trap" waves by making them bounce back and forth between two or more surfaces.

Musical instruments take advantage of this; they produce pitches (Section 1.1.3) by trapping sound

waves.

Why are trapped waves useful for music? Any bunch of sound waves will produce some sort of

tonenoise. But to be a - a sound with a particular pitch (Section 1.1.3) - a group of sound waves has

to be very regular, all exactly the same distance apart. That's why we can talk about the frequency

(pg 99) and wavelength (pg 99) of tones.

Figure 3.6: A noise is a jumble of sound waves. A tone is a very regular set of waves, all the

same size and same distance apart.

So how can you produce a tone? Let's say you have a sound wave trap (for now, don't worry
about what it looks like), and you keep sending more sound waves into it. Picture a lot of pebbles
being dropped into a very small pool. As the waves start re ecting o the edges of the pond, they
interfere with the new waves, making a jumble of waves that partly cancel each other out and mostly
just roils the pond - noise.

But what if you could arrange the waves so that re ecting waves, instead of cancelling out the
new waves, would reinforce them? The high parts of the re ected waves would meet the high parts
of the oncoming waves and make them even higher. The low parts of the re ected waves would meet
the low parts of the oncoming waves and make them even lower. Instead of a roiled mess of waves
cancelling each other out, you would have a pond of perfectly ordered waves, with high points and
low points appearing regularly at the same spots again and again. To help you imagine this, here

are animations of a single wave re ecting back and forth 8 and standing waves9.

This sort of orderliness is actually hard to get from water waves, but relatively easy to get
in sound waves, so that several completely di erent types of sound wave "containers" have been
developed into musical instruments. The two most common - strings and hollow tubes - will be
discussed below, but rst let's nish discussing what makes a good standing wave container, and
how this a ects music theory.

8http://cnx.org/content/m12413/latest/Re ectingWave.swf
9http://cnx.org/content/m12413/latest/WaterWaves.swf

102 CHAPTER 3. THE PHYSICAL BASIS

In order to get the necessary constant reinforcement, the container has to be the perfect size
(length) for a certain wavelength, so that waves bouncing back or being produced at each end
reinforce each other, instead of interfering with each other and cancelling each other out. And
it really helps to keep the container very narrow, so that you don't have to worry about waves
bouncing o the sides and complicating things. So you have a bunch of regularly-spaced waves that
are trapped, bouncing back and forth in a container that ts their wavelength perfectly. If you could
watch these waves, it would not even look as if they are traveling back and forth. Instead, waves
would seem to be appearing and disappearing regularly at exactly the same spots, so these trapped

waves are called standing waves.

Note: Although standing waves are harder to get in water, the phenomenon does ap-
parently happen very rarely in lakes, resulting in freak disasters. You can sometimes get
the same e ect by pushing a tub of water back and forth, but this is a messy experiment;
you'll know you are getting a standing wave when the water suddenly starts sloshing much
higher - right out of the tub!

For any narrow "container" of a particular length, there are plenty of possible standing waves

that don't t. But there are also many standing waves that do t. The longest wave that ts it is

fundamental rst harmoniccalled the . The next longest wave that ts is
. It is also called the

the second harmonic, or the rst overtone. The next longest wave is the third harmonic, or
second overtone, and so on.

Standing Wave Harmonics

Figure 3.7: There is a whole set of standing waves, called harmonics, that will t into any
"container" of a speci c length. This set of waves is called a harmonic series.

Notice that it doesn't matter what the length of the fundamental is; the waves in the second
harmonic must be half the length of the rst harmonic; that's the only way they'll both " t". The
waves of the third harmonic must be a third the length of the rst harmonic, and so on. This
has a direct e ect on the frequency and pitch of harmonics, and so it a ects the basics of music

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tremendously. To nd out more about these subjects, please see Frequency, Wavelength, and Pitch 10,
Harmonic Series11, or Musical Intervals, Frequency, and Ratio 12.

3.2.2 Standing Waves on Strings

You may have noticed an interesting thing in the animation (pg 101) of standing waves: there are

spots where the "water" goes up and down a great deal, and other spots where the "water level"

nodesdoesn't seem to move at all. All standing waves have places, called , where there is no wave

antinodesmotion, and , where the wave is largest. It is the placement of the nodes that determines

which wavelengths " t" into a musical instrument "container".

Nodes and Antinodes

Figure 3.8: As a standing wave waves back and forth (from the red to the blue position), there
are some spots called nodes that do not move at all; basically there is no change, no waving up-

and-down (or back-and-forth), at these spots. The spots at the biggest part of the wave - where

there is the most change during each wave - are called antinodes.

One "container" that works very well to produce standing waves is a thin, very taut string that
is held tightly in place at both ends. (There were some nice animations of waves on strings available

as of this writing at Musemath13.) Since the string is taut, it vibrates quickly, producing sound

waves, if you pluck it, or rub it with a bow. Since it is held tightly at both ends, that means there
has to be a node (pg 103) at each end of the string. Instruments that produce sound using strings

are called chordophones14, or simply strings15.

10"Frequency, Wavelength, and Pitch" <http://cnx.org/content/m11060/latest/>
11"Harmonic Series" <http://cnx.org/content/m11118/latest/>
12"Musical Intervals, Frequency, and Ratio" <http://cnx.org/content/m11808/latest/>
13http://www.musemath.com
14"Classifying Musical Instruments" <http://cnx.org/content/m11896/latest/#s21>
15"Orchestral Instruments" <http://cnx.org/content/m11897/latest/#s11>

104 CHAPTER 3. THE PHYSICAL BASIS

Standing Waves on a String

Figure 3.9: A string that's held very tightly at both ends can only vibrate at very particular

wavelengths. The whole string can vibrate back and forth. It can vibrate in halves, with a node at
the middle of the string as well as each end, or in thirds, fourths, and so on. But any wavelength
that doesn't have a node at each end of the string, can't make a standing wave on the string. To
get any of those other wavelengths, you need to change the length of the vibrating string. That is
what happens when the player holds the string down with a nger, changing the vibrating length
of the string and changing where the nodes are.

The fundamental (pg 102) wave is the one that gives a string its pitch (Section 1.1.3). But the

string is making all those other possible vibrations, too, all at the same time, so that the actual

vibration of the string is pretty complex. The other vibrations (the ones that basically divide the

harmonicsstring into halves, thirds and so on) produce a whole series of . We don't hear the

harmonics as separate notes, but we do hear them. They are what gives the string its rich, musical,

string-like sound - its timbre (Section 2.2). (The sound of a single frequency alone is a much more

mechanical, uninteresting, and unmusical sound.) To nd out more about harmonics and how they

a ect a musical sound, see Harmonic Series 16.

Exercise 3.1:

When the string player puts a nger down tightly on the string,

1.How has the part of the string that vibrates changed?

16"Harmonic Series" <http://cnx.org/content/m11118/latest/>

105

2.How does this change the sound waves that the string makes?
3.How does this change the sound that is heard?

(Solution to Exercise 3.1 on p. 114.)

3.2.3 Standing Waves in Wind Instruments

The string disturbs the air molecules around it as it vibrates, producing sound waves in the air. But

another great container for standing waves actually holds standing waves of air inside a long, narrow

tube. This type of instrument is called an aerophone 17, and the most well-known of this type of
instrument are often called wind instruments 18 because, although the instrument itself does vibrate

a little, most of the sound is produced by standing waves in the column of air inside the instrument.

If it is possible, have a reed player and a brass player demonstrate to you the sounds that their

mouthpieces make without the instrument. This will be a much "noisier" sound, with lots of extra

frequencies in it that don't sound very musical. But, when you put the mouthpiece on an instrument

shaped like a tube, only some of the sounds the mouthpiece makes are the right length for the tube.

Because of feedback from the instrument, the only sound waves that the mouthpiece can produce

standing wavesnow are the ones that are just the right length to become in the instrument, and

the "noise" is re ned into a musical tone.

17"Classifying Musical Instruments" <http://cnx.org/content/m11896/latest/#s22>
18"Orchestral Instruments" <http://cnx.org/content/m11897/latest/#s1>

106 CHAPTER 3. THE PHYSICAL BASIS

Standing Waves in Wind Instruments

Figure 3.10: Standing Waves in a wind instrument are usually shown as displacement waves,

with nodes at closed ends where the air cannot move back-and-forth.

The standing waves in a wind instrument are a little di erent from a vibrating string. The wave

transverse waveon a string is a , moving the string back and forth, rather than moving up and

longitudinaldown along the string. But the wave inside a tube, since it is a sound wave already, is a
wave; the waves do not go from side to side in the tube. Instead, they form along the length of the

tube.

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Longitudinal Waves in Pipes

Figure 3.11: The standing waves in the tubes are actually longitudinal sound waves. Here the

displacement standing waves in Figure 3.10 are shown instead as longitudinal air pressure waves.
Each wave would be oscillating back and forth between the state on the right and the one on the
left. See Standing Waves in Wind Instruments19 for more explanation.

The harmonics of wind instruments are also a little more complicated, since there are two basic

shapes (cylindrical20 and conical21) that are useful for wind instruments, and they have di erent

properties. The standing-wave tube of a wind instrument also may be open at both ends, or it
may be closed at one end (for a mouthpiece, for example), and this also a ects the instrument.

Please see Standing Waves in Wind Instruments 22 if you want more information on that subject.

For the purposes of understanding music theory, however, the important thing about standing waves
in winds is this: the harmonic series they produce is essentially the same as the harmonic series on
a string. In other words, the second harmonic is still half the length of the fundamental, the third
harmonic is one third the length, and so on. (Actually, for reasons explained in Standing Waves

in Wind Instruments23, some harmonics are "missing" in some wind instruments, but this mainly

a ects the timbre (Section 2.2) and some aspects of playing the instrument. It does not a ect the
basic relationships in the harmonic series.)

19"Standing Waves and Wind Instruments" <http://cnx.org/content/m12589/latest/>
20"Wind Instruments: Some Basics" <http://cnx.org/content/m12364/latest/#p1c>
21"Wind Instruments: Some Basics" <http://cnx.org/content/m12364/latest/#p1c>
22"Standing Waves and Wind Instruments" <http://cnx.org/content/m12589/latest/>
23"Standing Waves and Wind Instruments" <http://cnx.org/content/m12589/latest/>

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3.2.4 Standing Waves in Other Objects

So far we have looked at two of the four main groups of musical instruments: chordophones and

Membranophonesaerophones. That leaves membranophones24 and idiophones25. are instruments

in which the sound is produced by making a membrane vibrate; drums are the most familiar example.

Most drums do not produce tones; they produce rhythmic "noise" (bursts of irregular waves). Some

drums do have pitch (Section 1.1.3), due to complex-patterned standing waves on the membrane that

are reinforced in the space inside the drum. This works a little bit like the waves in tubes, above,

but the waves produced on membranes, though very interesting, are too complex to be discussed

here.

Idiophones are instruments in which the body of the instrument itself, or a part of it, produces

the original vibration. Some of these instruments (cymbals, for example) produce simple noise-

like sounds when struck. But in some, the shape of the instrument - usually a tube, block, circle,

or bell shape - allows the instrument to ring with a standing-wave vibration when you strike it.

The standing waves in these carefully-shaped-and-sized idiophones - for example, the blocks on a

xylophone - produce pitched tones, but again, the patterns of standing waves in these instruments

are a little too complicated for this discussion. If a percussion instrument does produce pitched

sounds, however, the reason, again, is that it is mainly producing harmonic-series overtones 26.

Note: Although percussion27 specializes in "noise"-type sounds, even instruments like

snare drums follow the basic physics rule of "bigger instrument makes longer wavelengths
and lower sounds". If you can, listen to a percussion player or section that is using snare
drums, cymbals, or other percussion of the same type but di erent sizes. Can you hear the
di erence that size makes, as opposed to di erences in timbre (Section 2.2) produced by
di erent types of drums?

Exercise 3.2:

Some idiophones, like gongs, ring at many di erent pitches when they are struck. Like most

drums, they don't have a particular pitch, but make more of a "noise"-type sound. Other

idiophones, though, like xylophones, are designed to ring at more particular frequencies.

Can you think of some other percussion instruments that get particular pitches? (Some

can get enough di erent pitches to play a tune.) (Solution to Exercise 3.2 on p. 114.)

3.3 Harmonic Series I: Timbre and Octaves28

3.3.1 Introduction

Have you ever wondered how a trumpet29 plays so many di erent notes with only three valves 30, or
how a bugle plays di erent notes with no valves at all? Have you ever wondered why an oboe 31 and
a ute32 sound so di erent, even when they're playing the same note? What is a string player doing

when she plays "harmonics"? Why do some notes sound good together while other notes seem to
clash with each other? The answers to all of these questions have to do with the harmonic series.

24"Classifying Musical Instruments" <http://cnx.org/content/m11896/latest/#s23>
25"Classifying Musical Instruments" <http://cnx.org/content/m11896/latest/#s24>
26"Harmonic Series" <http://cnx.org/content/m11118/latest/>
27"Orchestral Instruments" <http://cnx.org/content/m11897/latest/#s14>
28This content is available online at <http://cnx.org/content/m13682/1.5/>.
29"Trumpets and Cornets" <http://cnx.org/content/m12606/latest/>
30"Wind Instruments: Some Basics" <http://cnx.org/content/m12364/latest/#p2f>
31"The Oboe and its Relatives" <http://cnx.org/content/m12615/latest/>
32"Flutes" <http://cnx.org/content/m12603/latest/>

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3.3.2 Physics, Harmonics and Color

Most musical notes are sounds that have a particular pitch (Section 1.1.3). The pitch depends on the
main frequency (Section 3.1.4) of the sound; the higher the frequency, and shorter the wavelength
(Section 3.1.4), of the sound waves, the higher the pitch is. But musical sounds don't have just one
frequency. Sounds that have only one frequency are not very interesting or pretty. They have no
more musical color (Section 2.2) than the beeping of a watch alarm. On the other hand, sounds that
have too many frequencies, like the sound of glass breaking or of ocean waves crashing on a beach,
may be interesting and even pleasant. But they don't have a particular pitch, so they usually aren't
considered musical notes.

Frequency and Pitch

Figure 3.12: The shorter the wavelength, and higher the frequency, the higher the note sounds.

When someone plays or sings a musical tone (pg 101), only a very particular set of frequencies is

heard. Each note that comes out of the instrument is actually a smooth mixture of many di erent

harmonicspitches. These di erent pitches are called , and they are blended together so well that

you do not hear them as separate notes at all. Instead, the harmonics give the note its color.

What is the color (Section 2.2) of a sound? Say an oboe plays a middle C (pg 120). Then a ute

plays the same note at the same dynamic (Section 1.3.1) level as the oboe. It is still easy to tell

the two notes apart, because an oboe sounds di erent from a ute. This di erence in the sounds

color timbreis the
, or (pronounced "TAM-ber") of the notes. Like a color you see, the color of a

sound can be bright and bold or deep and rich. It can be heavy, light, dark, thin, smooth, murky,

or clear. Some other words that musicians use to describe the timbre of a sound are: reedy, brassy,

piercing, mellow, hollow, focussed, transparent, breathy (pronounced BRETH-ee) or full. Listen to

recordings of a violin33 and a viola34. Although these instruments are quite similar, the viola has

a noticeably "deeper" and the violin a noticeably "brighter" sound that is not simply a matter of

33http://cnx.org/content/m13682/latest/timvl.mp3
34http://cnx.org/content/m13682/latest/timvla.mp3

110 CHAPTER 3. THE PHYSICAL BASIS

the violin playing higher notes. Now listen to the same phrase played by an electric guitar 35, an
acoustic guitar with twelve steel strings36 and an acoustic guitar with six nylon strings 37. The words

musicians use to describe timbre are somewhat subjective, but most musicians would agree with the
statement that, compared with each other, the rst sound is mellow, the second bright, and the
third rich.

Exercise 3.3:

Listen to recordings of di erent instruments playing alone or playing very prominently
above a group. Some suggestions: an unaccompanied violin or cello sonata, a ute, oboe,
trumpet, or horn concerto, Asaian or native American ute music, classical guitar, bag-
pipes, steel pan drums, panpipes, or organ. For each instrument, what "color" words would
you use to describe the timbre of each instrument? Use as many words as you can that
seem appropriate, and try to think of some that aren't listed above. Do any of the instru-
ments actually make you think of speci c shades of color, like re-engine red or sky blue?

(Solution to Exercise 3.3 on p. 114.)

Where do the harmonics, and the timbre, come from? When a string vibrates, the main pitch

fundamentalyou hear is from the vibration of the whole string back and forth. That is the , or

rst harmonic. But the string also vibrates in halves, in thirds, fourths, and so on. (Please see

Standing Waves and Musical Instruments (Section 3.2) for more on the physics of how harmonics

are produced.) Each of these fractions also produces a harmonic. The string vibrating in halves

produces the second harmonic; vibrating in thirds produces the third harmonic, and so on.

Note: This method of naming and numbering harmonics is the most straightforward and

least confusing, but there are other ways of naming and numbering harmonics, and this

can cause confusion. Some musicians do not consider the fundamental to be a harmonic;

it is just the fundamental. In that case, the string halves will give the rst harmonic, the

string thirds will give the second harmonic and so on. When the fundamental is included in

partialcalculations, it is called the rst , and the rest of the harmonics are the second, third,

overtonesfourth partials and so on. Also, some musicians use the term as a synonym for

harmonics. For others, however, an overtone is any frequency (not necessarily a harmonic)

that can be heard resonating with the fundamental. The sound of a gong or cymbals will

include overtones that aren't harmonics; that's why the gong's sound doesn't seem to have

as de nite a pitch as the vibrating string does. If you are uncertain what someone means

when they refer to "the second harmonic" or "overtones", ask for clari cation.

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Vibrating String

Figure 3.13: The fundamental pitch is produced by the whole string vibrating back and forth.
But the string is also vibrating in halves, thirds, quarters, fths, and so on, producing harmonics.

All of these vibrations happen at the same time, producing a rich, complex, interesting sound.

A column of air vibrating inside a tube is di erent from a vibrating string, but the column of
air can also vibrate in halves, thirds, fourths, and so on, of the fundamental, so the harmonic series
will be the same. So why do di erent instruments have di erent timbres? The di erence is the

relative loudness of all the di erent harmonics compared to each other. When a clarinet 38 plays
a note, perhaps the odd-numbered harmonics are strongest; when a French horn 39 plays the same

note, perhaps the fth and tenth harmonics are the strongest. This is what you hear that allows
you to recognize that it is a clarinet or horn that is playing. The relative strength of the harmonics
changes from note to note on the same instrument, too; this is the di erence you hear between the
sound of a clarinet playing low notes and the same clarinet playing high notes.

Note: You will nd some more extensive information on instruments and harmonics
in Standing Waves and Musical Instruments (Section 3.2) and Standing Waves and Wind

Instruments40 .

3.3.3 The Harmonic Series

A harmonic series can have any note as its fundamental, so there are many di erent harmonic series.
But the relationship between the frequencies of a harmonic series is always the same. The second
harmonic always has exactly half the wavelength (and twice the frequency) of the fundamental; the
third harmonic always has exactly a third of the wavelength (and so three times the frequency) of
the fundamental, and so on. For more discussion of wavelengths and frequencies, see Acoustics for
Music Theory (Section 3.1).

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112 CHAPTER 3. THE PHYSICAL BASIS

Harmonic Series Wavelengths and Frequencies

Figure 3.14: The second harmonic has half the wavelength and twice the frequency of the rst.

The third harmonic has a third the wavelength and three times the frequency of the rst. The
fourth harmonic has a quarter the wavelength and four times the frequency of the rst, and so on.
Notice that the fourth harmonic is also twice the frequency of the second harmonic, and the sixth
harmonic is also twice the frequency of the third harmonic.

Say someone plays a note, a middle C (pg 120). Now someone else plays the note that is twice
the frequency of the middle C. Since this second note was already a harmonic of the rst note, the
sound waves of the two notes reinforce each other and sound good together. If the second person
played instead the note that was just a litle bit more than twice the frequency of the rst note, the
harmonic series of the two notes would not t together at all, and the two notes would not sound
as good together. There are many combinations of notes that share some harmonics and make a
pleasant sound together. They are considered consonant (Section 5.3). Other combinations share
fewer or no harmonics and are considered dissonant (Section 5.3) or, when they really clash, simply
"out of tune" with each other. The scales (Section 4.8) and harmonies (Section 2.5) of most of the
world's musics are based on these physical facts.

Note: In real music, consonance and dissonance also depend on the standard practices
of a musical tradition, especially its harmony and tuning (Section 6.2) practices, but these
are also often related to the harmonic series.
For example, a note that is twice the frequency of another note is one octave (Section 4.1) higher
than the rst note. So in the gure above, the second harmonic is one octave higher than the rst;
the fourth harmonic is one octave higher than the second; and the sixth harmonic is one octave

113

higher than the third.

Exercise 3.4:

1.Which harmonic will be one octave higher than the fourth harmonic?
2.Predict the next four sets of octaves in a harmonic series.
3.What is the pattern that predicts which notes of a harmonic series will be one octave

apart?
4.Notes one octave apart are given the same name. So if the rst harmonic is a "A",

the second and fourth will also be A's. Name three other harmonics that will also be
A's.

(Solution to Exercise 3.4 on p. 114.)

A mathematical way to say this is "if two notes are an octave apart, the ratio 41 of their frequencies

is two to one (2:1)". Although the notes themselves can be any frequency, the 2:1 ratio is the same
for all octaves. Other frequency ratios between two notes also lead to particular pitch relationships
between the notes, so we will return to the harmonic series later, after learning to name those pitch
relationships, or intervals (Section 4.5).

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114 CHAPTER 3. THE PHYSICAL BASIS

Solutions to Exercises in Chapter 3

Solution to Exercise 3.1 (p. 104):

1. The part of the string that can vibrate is shorter. The nger becomes the new "end" of the
string.

2. The new sound wave is shorter, so its frequency is higher.
3. It sounds higher; it has a higher pitch.

Figure 3.15: When a nger holds the string down tightly, the nger becomes the new end of the

vibrating part of the string. The vibrating part of the string is shorter, and the whole set of sound
waves it makes is shorter.

Solution to Exercise 3.2 (p. 108):
There are many, but here are some of the most familiar:

• Chimes
• All xylophone-type instruments, such as marimba, vibraphone, and glockenspiel
• Handbells and other tuned bells
• Steel pan drums

Solution to Exercise 3.3 (p. 110):

Although trained musicians will generally agree that a particular sound is reedy, thin, or full,
there are no hard-and-fast, right-or-wrong answers to this exercise.

Solution to Exercise 3.4 (p. 113):

1. The eighth harmonic
2. The fth and tenth harmonics; the sixth and twelfth harmonics; the seventh and fourteenth

harmonics; and the eighth and sixteenth harmonics

115

3. The note that is one octave higher than a harmonic is also a harmonic, and its number in the
harmonic series is twice (2 X) the number of the rst note.

4. The eighth, sixteenth, and thirty-second harmonics will also be A's.

116 CHAPTER 3. THE PHYSICAL BASIS

Chapter 4

Notes and Scales

4.1 Octaves and the Ma jor-Minor Tonal System1

4.1.1 Where Octaves Come From

Musical notes, like all sounds, are made of sound waves. The sound waves that make musical notes

are very evenly-spaced waves, and the qualities of these regular waves - for example how big they

are or how far apart they are - a ect the sound of the note. A note can be high or low, depending on

how often (how frequently) one of its waves arrives at your ear. When scientists and engineers talk

frequencyabout how high or low a sound is, they talk about its frequency 2. The higher the of a

note, the higher it sounds. They can measure the frequency of notes, and like most measurements,

these will be numbers, like "440 vibrations per second."

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118 CHAPTER 4. NOTES AND SCALES

High and Low Frequencies

Figure 4.1: A sound that has a shorter wavelength has a higher frequency and a higher pitch.

But people have been making music and talking about music since long before we knew that
sounds were waves with frequencies. So when musicians talk about how high or low a note sounds,
they usually don't talk about frequency; they talk about the note's pitch (Section 1.1.3). And
instead of numbers, they give the notes names, like "C". (For example, musicians call the note with
frequency "440 vibrations per second" an "A".)

But to see where octaves come from, let's talk about frequencies a little more. Imagine a few
men are singing a song together. Nobody is singing harmony; they are all singing the same pitch -
the same frequency - for each note.

Now some women join in the song. They can't sing where the men are singing; that's too low for
their voices. Instead they sing notes that are exactly double the frequency that the men are singing.
That means their note has exactly two waves for each one wave that the men's note has. These two
frequencies t so well together that it sounds like the women are singing the same notes as the men,

Any note that is twicein the same key (Section 4.3). They are just singing them one octave higher.
the frequency of another note is one octave higher.

Notes that are one octave apart are so closely related to each other that musicians give them the
same name. A note that is an octave higher or lower than a note named "C natural" will also be
named "C natural". A note that is one (or more) octaves higher or lower than an "F sharp" will
also be an "F sharp". (For more discussion of how notes are related because of their frequencies, see

The Harmonic Series3, Standing Waves and Musical Instruments (Section 3.2), and Standing Waves
and Wind Instruments4.)

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Octave Frequencies

Figure 4.2: When two notes are one octave apart, one has a frequency exactly two times higher

than the other - it has twice as many waves. These waves t together so well, in the instrument,
and in the air, and in your ears, that they sound almost like di erent versions of the same note.

4.1.2 Naming Octaves

The notes in di erent octaves are so closely related that when musicians talk about a note, a "G"
for example, it often doesn't matter which G they are talking about. We can talk about the "F
sharp" in a G major scale (Section 4.3) without mentioning which octave the scale or the F sharp
are in, because the scale is the same in every octave. Because of this, many discussions of music
theory don't bother naming octaves. Informally, musicians often speak of "the B on the sta " or
the "A above the sta ", if it's clear which sta (Section 1.1.1) they're talking about.

But there are also two formal systems for naming the notes in a particular octave. Many musicians

use Helmholtz notation. Others prefer scienti c pitch notation, which simply labels the octaves

with numbers, starting with C1 for the lowest C on a full-sized keyboard. Figure 3 shows the names
of the octaves most commonly used in music.

120 CHAPTER 4. NOTES AND SCALES

Naming Octaves

Figure 4.3: The octaves are named from one C to the next higher C. For example, all the notes

in between "one line c" and "two line c" are "one line" notes.

The octave below contra can be labelled CCC or Co; higher octaves can be labelled with higher
numbers or more lines. Octaves are named from one C to the next higher C. For example, all the

notes between "great C" and "small C" are "great". One-line c is also often called "middle C". No
other notes are called "middle", only the C.

Example 4.1:
Naming Notes within a Particular Octave

Figure 4.4: Each note is considered to be in the same octave as the C below it.

121

Exercise 4.1:

Give the correct octave name for each note.

Figure 4.5

(Solution to Exercise 4.1 on p. 162.)

4.1.3 Dividing the Octave into Scales

The word "octave" comes from a Latin root meaning "eight". It seems an odd name for a frequency

that is two times, not eight times, higher. The octave was named by musicians who were more

interested in how octaves are divided into scales, than in how their frequencies are related. Octaves

aren't the only notes that sound good together. The people in di erent musical traditions have

di erent ideas about what notes they think sound best together. In the Western (Section 2.8)

musical tradition - which includes most familiar music from Europe and the Americas - the octave is

divided up into twelve equally spaced notes. If you play all twelve of these notes within one octave

you are playing a chromatic scale (pg 123). Other musical traditions - traditional Chinese music

for example - have divided the octave di erently and so they use di erent scales. (Please see Major

Keys and Scales (Section 4.3), Minor Keys and Scales (Section 4.4), and Scales that aren't Major

or Minor (Section 4.8) for more about this.)

You may be thinking "OK, that's twelve notes; that still has nothing to do with the number

eight", but out of those twelve notes, only seven are used in any particular major (Section 4.3) or

minor (Section 4.4) scale. Add the rst note of the next octave, so that you have that a "complete"-

sounding scale ("do-re-mi-fa-so-la-ti" and then "do" again), and you have the eight notes of the

octave diatonic. These are the scales, and they are the basis of most Western (Section 2.8) music.

Now take a look at the piano keyboard. Only seven letter names are used to name notes: A, B, C,

D, E, F, and G. The eighth note would, of course, be the next A, beginning the next octave. To name

the other notes, the notes on the black piano keys, you have to use a sharp or at (Section 1.1.3)

sign.

122 CHAPTER 4. NOTES AND SCALES

Keyboard

Figure 4.6: The white keys are the natural notes. Black keys can only be named using sharps

or ats. The pattern repeats at the eighth tone of a scale, the octave.

Whether it is a popular song, a classical symphony, or an old folk tune, most of the music that

feels comfortable and familiar (to Western listeners) is based on either a major or minor scale. It is

tonal music that mostly uses only seven of the notes within an octave: only one of the possible A's

(A sharp, A natural, or A at), one of the possible B's (B sharp, B natural, or B at), and so on. The

other notes in the chromatic scale are (usually) used sparingly to add interest or to (temporarily)

change the key in the middle of the music. For more on the keys and scales that are the basis of

tonal music, see Major Keys and Scales (Section 4.3) and Minor Keys and Scales (Section 4.4).

4.2 Half Steps and Whole Steps5

pitchThe of a note is how high or low it sounds. Musicians often nd it useful to talk about how

much higher or lower one note is than another. This distance between two pitches is called the

interval between them. In Western music (Section 2.8), the small interval from one note to the
half step semi-tonenext closest note higher or lower is called a
or .

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123

Half Steps

(a)

(b)

Figure 4.7: Three half-step intervals: between C and C sharp (or D at); between E and F; and

between G sharp (or A at) and A.

Listen6 to the half steps in Figure 4.7.

The intervals in Figure 4.7 look di erent on a sta (Section 1.1.1); sometimes they are on the

same line, sometimes not. But it is clear at the keyboard that in each case there is no note in

between them.

chromatic scaleSo a scale (Section 4.3) that goes up or down by half steps, a , plays all the

notes on both the white and black keys of a piano. It also plays all the notes easily available on

most Western (Section 2.8) instruments. (A few instruments, like trombone 7 and violin8, can easily

play pitches that aren't in the chromatic scale, but even they usually don't.)

One Octave Chromatic Scale

Figure 4.8: All intervals in a chromatic scale are half steps. The result is a scale that plays all

the notes easily available on most instruments.
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124 CHAPTER 4. NOTES AND SCALES

Listen9 to a chromatic scale.

wholeIf you go up or down two half steps from one note to another, then those notes are a
step, or whole tone apart.

Whole Steps

(a)

(b)

Figure 4.9: Three whole step intervals: between C and D; between E and F sharp; and between

G sharp and A sharp (or A at and B at).

whole tone scaleA , a scale made only of whole steps, sounds very di erent from a chromatic

scale.

Whole Tone Scale

Figure 4.10: All intervals in a whole tone scale are whole steps.

Listen10 to a whole tone scale.

You can count any number of whole steps or half steps between notes; just remember to count
all sharp or at notes (the black keys on a keyboard) as well as all the natural notes (the white keys)
that are in between.

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125

Example 4.2:

The interval between C and the F above it is 5 half steps, or two and a half steps.

Figure 4.11: Going from C up to F takes ve half steps.

Exercise 4.2:

Identify the intervals below in terms of half steps and whole steps. If you have trouble
keeping track of the notes, use a piano keyboard, a written chromatic scale, or the chromatic
ngerings for your instrument to count half steps.

Figure 4.12

(Solution to Exercise 4.2 on p. 162.)

126 CHAPTER 4. NOTES AND SCALES

Exercise 4.3:

Fill in the second note of the interval indicated in each measure. If you need sta paper

for this exercise, you can print out this sta paper 11 PDF le.

Figure 4.13

(Solution to Exercise 4.3 on p. 163.)

4.3 Ma jor Keys and Scales12

The simple, sing-along, nursery rhymes and folk songs we learn as children, the cheerful, toe-tapping

pop and rock we dance to, the uplifting sounds of a symphony: most music in a major key has a

bright sound that people often describe as cheerful, inspiring, exciting, or just plain fun.

keyMusic in a particular tends to use only some of the many possible notes available; these

scalenotes are listed in the associated with that key. The notes that a major key uses tend to build

"bright"-sounding major chords. They also give a strong feeling of having a tonal center (pg 127),

a note or chord that feels like "home" in that key. The "bright"-sounding major chords and the

strong feeling of tonality are what give major keys their pleasant moods.

Exercise 4.4:

Listen to these excerpts. Three are in a major key and two in a minor key. Can you tell
which is which simply by listening?

• 1.13
• 2.14
• 3.15
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127

• 4.16
• 5.17

(Solution to Exercise 4.4 on p. 163.)

4.3.1 Tonal Center

tonal centerA scale starts with the note that names the key. This note is the of that key, the note

tonicwhere music in that key feels "at rest". It is also called the , and it's the "do" in "do-re-mi".

For example, music in the key of A major almost always ends on an A major chord, the chord

(Chords) built on the note A. It often also begins on that chord, returns to that chord often, and

features a melody and a bass line that also return to the note A often enough that listeners will

know where the tonal center of the music is, even if they don't realize that they know it. (For more

information about the tonic chord and its relationship to other chords in a key, please see Beginning

Harmonic Analysis (Section 5.5).)

Example 4.3:

Listen to these examples. Can you hear that they do not feel "done" until the nal tonic
is played?

• Example A18
• Example B19

4.3.2 Major Scales

To nd the rest of the notes in a major key, start at the tonic and go up following this pattern:

whole step, whole step, half step, whole step, whole step, whole step, half step . This will take you

to the tonic one octave higher than where you began, and includes all the notes in the key in that
octave.

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128 CHAPTER 4. NOTES AND SCALES

Example 4.4:

These major scales all follow the same pattern of whole steps and half steps. They have
di erent sets of notes because the pattern starts on di erent notes.

Three Major Scales

Figure 4.14: All major scales have the same pattern of half steps and whole steps, beginning on

the note that names the scale - the tonic (pg 127).

Listen to the di erence between the C major 20, D major21, and B at major22 scales.

Exercise 4.5:

For each note below, write a major scale, one octave, ascending (going up), beginning on
that note. If you're not sure whether a note should be written as a at, sharp, or natural,
remember that you won't ever skip a line or space, or write two notes of the scale on the
same line or space. If you need help keeping track of half steps, use a keyboard, a picture
of a keyboard (Figure 4.6), a written chromatic scale (pg 123), or the chromatic scale
ngerings for your instrument. If you need more information about half steps and whole
steps, see Half Steps and Whole Steps (Section 4.2).

If you need sta paper for this exercise, you can print out this sta paper 23 PDF le.

20http://cnx.org/content/m10851/latest/tonmjC.mp3
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129

Figure 4.15

(Solution to Exercise 4.5 on p. 164.)

In the examples above, the sharps and ats are written next to the notes. In common notation,

keythe sharps and ats that belong in the key will be written at the beginning of each sta , in the
signature. For more practice identifying keys and writing key signatures, please see Key Signature

(Section 1.1.4). For more information about how keys are related to each other, please see The
Circle of Fifths (Section 4.7).

4.3.3 Music in Di erent Keys

What di erence does key make? Since the major scales all follow the same pattern, they all sound
very much alike. Here is a folk tune ("The Saucy Sailor") written in D major and in F major.

130 CHAPTER 4. NOTES AND SCALES

(a)

(b)

Figure 4.16: The same tune looks very di erent written in two di erent major keys.

Listen to this tune in D major24 and in F major25. The music may look quite di erent, but

the only di erence when you listen is that one sounds higher than the other. So why bother with
di erent keys at all? Before equal temperament (Section 6.2.3.2) became the standard tuning system,
major keys sounded more di erent from each other than they do now. Even now, there are subtle
di erences between the sound of a piece in one key or another, mostly because of di erences in the
timbre (Section 2.2) of various notes on the instruments or voices involved. But today the most
common reason to choose a particular key is simply that the music is easiest to sing or play in that
key. (Please see Transposition (Section 6.4) for more about choosing keys.)

4.4 Minor Keys and Scales26

4.4.1 Music in a Minor Key

Each major key (Section 4.3) uses a di erent set of notes (Section 1.2.1) (its major scale (Sec-
tion 4.3.2)). In each major scale, however, the notes are arranged in the same major scale pattern
and build the same types of chords that have the same relationships with each other. (See Beginning
Harmonic Analysis (Section 5.5) for more on this.) So music that is in, for example, C major, will
not sound signi cantly di erent from music that is in, say, D major. But music that is in D minor
will have a di erent quality, because the notes in the minor scale follow a di erent pattern and so
have di erent relationships with each other. Music in minor keys has a di erent sound and emo-
tional feel, and develops di erently harmonically. So you can't, for example, transpose (Section 6.4)

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131

a piece from C major to D minor (or even to C minor) without changing it a great deal. Music
that is in a minor key is sometimes described as sounding more solemn, sad, mysterious, or ominous
than music that is in a major key. To hear some simple examples in both major and minor keys,
see Major Keys and Scales (Exercise 4.4).

4.4.2 Minor Scales

Minor scales sound di erent from major scales because they are based on a di erent pattern of

intervals (Section 4.5). Just as it did in major scales, starting the minor scale pattern on a di erent

note will give you a di erent key signature (Section 1.1.4), a di erent set of sharps or ats. The

natural minor scalescale that is created by playing all the notes in a minor key signature is a . To

create a natural minor scale, start on the tonic note (pg 127) and go up the scale using the interval

pattern: whole step, half step, whole step, whole step, half step, whole step, whole step .

Natural Minor Scale Intervals

Figure 4.17

Listen27 to these minor scales.

Exercise 4.6:

For each note below, write a natural minor scale, one octave, ascending (going up) beginning

on that note. If you need sta paper, you may print the sta paper 28 PDF le.

27http://cnx.org/content/m10856/latest/3a.mid
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132 CHAPTER 4. NOTES AND SCALES

Figure 4.18

(Solution to Exercise 4.6 on p. 165.)

4.4.3 Relative Minor and Major Keys

Each minor key shares a key signature (Section 1.1.4) with a major key. A minor key is called the

relative minor of the major key that has the same key signature. Even though they have the same
relative majorkey signature, a minor key and its
sound very di erent. They have di erent tonal

centers (pg 127), and each will feature melodies, harmonies, and chord progressions (Chords) built

around their (di erent) tonal centers. In fact, certain strategic accidentals (pg 17) are very useful in

helping establish a strong tonal center in a minor key. These useful accidentals are featured in the

melodic minor (Section 4.4.3) and harmonic minor (Section 4.4.3) scales.

Comparing Major and Minor Scale Patterns

Figure 4.19: The interval patterns for major and natural minor scales are basically the same

pattern starting at di erent points.

It is easy to predict where the relative minor of a major key can be found. Notice that the
pattern for minor scales overlaps the pattern for major scales. In other words, they are the same
pattern starting in a di erent place. (If the patterns were very di erent, minor key signatures would
not be the same as major key signatures.) The pattern for the minor scale starts a half step plus a

whole step lower than the major scale pattern, so a relative minor is always three half steps lower
than its relative major. For example, C minor has the same key signature as E at major, since E

at is a minor third higher than C.

133

Relative Minor

Figure 4.20: The C major and C minor scales start on the same note, but have di erent key

signatures. C minor and E at major start on di erent notes, but have the same key signature. C

minor is the relative minor of E at major.

Exercise 4.7:

What are the relative majors of the minor keys in Figure 4.18? (Solution to Exercise 4.7 on
p. 166.)

4.4.4 Harmonic and Melodic Minor Scales

natural minor scalesAll of the scales above are . They contain only the notes in the minor key

signature. There are two other kinds of minor scales that are commonly used, both of which include

notes that are not in the key signature. The harmonic minor scale raises the seventh note of the

scale by one half step, whether you are going up or down the scale . Harmonies in minor keys often

use this raised seventh tone in order to make the music feel more strongly centered on the tonic (pg

melodic127). (Please see Beginning Harmonic Analysis (Section 5.5.5) for more about this.) In the
minor scale, the sixth and seventh notes of the scale are each raised by one half step when going

up the scale, but return to the natural minor when going down the scale . Melodies in minor keys

often use this particular pattern of accidentals (pg 17), so instrumentalists nd it useful to practice

melodic minor scales.

134 CHAPTER 4. NOTES AND SCALES

Comparing Types of Minor Scales

Figure 4.21

Listen to the di erences between the natural minor 29, harmonic minor30, and melodic minor31

scales.

Exercise 4.8:

Rewrite each scale from Figure 4.18 as an ascending harmonic minor scale. (Solution to

Exercise 4.8 on p. 166.)

Exercise 4.9:

Rewrite each scale from Figure 4.18 as an ascending and descending melodic minor scale.

(Solution to Exercise 4.9 on p. 167.)

4.4.5 Jazz and "Dorian Minor"

Major and minor scales are traditionally the basis for Western Music (Section 2.8), but jazz theory

also recognizes other scales, based on the medieval church modes (pg 236), which are very useful for

improvisation. One of the most useful of these is the scale based on the dorian mode, which is often

dorian minorcalled the , since it has a basically minor sound. Like any minor scale, dorian minor

may start on any note, but like dorian mode, it is often illustrated as natural notes beginning on d.

29http://cnx.org/content/m10856/latest/tonminnatural.mp3
30http://cnx.org/content/m10856/latest/tonminharmonic.mp3
31http://cnx.org/content/m10856/latest/tonminmelodic.mp3

135

Dorian Minor

Figure 4.22: The "dorian minor" can be written as a scale of natural notes starting on d. Any

scale with this interval pattern can be called a "dorian minor scale".

Comparing this scale to the natural minor scale makes it easy to see why the dorian mode sounds
minor; only one note is di erent.

Comparing Dorian and Natural Minors

Figure 4.23

You may nd it helpful to notice that the "relative major" of the Dorian begins one whole step

lower. (So, for example, D Dorian has the same key signature as C major.) In fact, the reason that

Dorian is so useful in jazz is that it is the scale used for improvising while a ii chord (Section 5.5.2)

is being played (for example, while a d minor chord is played in the key of C major), a chord which

is very common in jazz. (See Beginning Harmonic Analysis (Section 5.5) for more about how chords

are classi ed within a key.) The student who is interested in modal jazz will eventually become

modal scalesacquainted with all of the . Each of these is named for the medieval church mode (pg

236) which has the same interval pattern, and each can be used with a di erent chord within the

key. Dorian is included here only to explain the common jazz reference to the "dorian minor" and

to give notice to students that the jazz approach to scales can be quite di erent from the traditional

classical approach.

136 CHAPTER 4. NOTES AND SCALES

Comparison of Dorian and Minor Scales

Figure 4.24: You may also nd it useful to compare the dorian with the minor scales from

Figure 4.21. Notice in particular the relationship of the altered notes in the harmonic, melodic,
and dorian minors.

4.5 Interval32

4.5.1 The Distance Between Pitches

intervalThe between two notes is the distance between the two pitches (Section 1.1.3) - in other

words, how much higher or lower one note is than the other. This concept is so important that

it is almost impossible to talk about scales (Section 4.3), chords (Chords), harmonic progression

(Chords), cadence (Section 5.6), or dissonance (Section 5.3) without referring to intervals. So if you

want to learn music theory, it would be a good idea to spend some time getting comfortable with

the concepts below and practicing identifying intervals.

Scientists usually describe the distance between two pitches in terms of the di erence between

their frequencies33. Musicians nd it more useful to talk about interval. Intervals can be described

using half steps and whole steps (Section 4.2). For example, you can say "B natural is a half step

below C natural", or "E at is a step and a half above C natural". But when we talk about larger

intervals in the major/minor system (Section 4.1), there is a more convenient and descriptive way

to name them.

32This content is available online at <http://cnx.org/content/m10867/2.21/>.
33"Frequency, Wavelength, and Pitch" <http://cnx.org/content/m11060/latest/>

137

4.5.2 Naming Intervals

as they are writtenThe rst step in naming the interval is to nd the distance between the notes
on the sta . Count every line and every space in between the notes, as well as the lines or spaces

that the notes are on. This gives you the number for the interval.

Example 4.5:
Counting Intervals

Figure 4.25

To nd the interval, count the lines or spaces that the two notes are on as well as all the
lines or spaces in between. The interval between B and D is a third. The interval between A
and F is a sixth. Note that, at this stage, key signature (Section 1.1.4), clef (Section 1.1.2),
and accidentals (pg 17) do not matter at all.

The simple intervals are one octave or smaller.
Simple Intervals

Figure 4.26

138 CHAPTER 4. NOTES AND SCALES

If you like you can listen to each interval as written in Figure 4.26: prime 34, second35, third36,
fourth37, fth38, sixth39, seventh40, octave41.

Compound intervals are larger than an octave.

Compound Intervals

Figure 4.27

Listen to the compound intervals in Figure 4.27: ninth 42, tenth43, eleventh44.

Exercise 4.10:

Name the intervals.

Figure 4.28

(Solution to Exercise 4.10 on p. 167.)

Exercise 4.11:

Write a note that will give the named interval.

34http://cnx.org/content/m10867/latest/prime.mid
35http://cnx.org/content/m10867/latest/second.mid
36http://cnx.org/content/m10867/latest/third.mid
37http://cnx.org/content/m10867/latest/fourht.mid
38http://cnx.org/content/m10867/latest/ fth.mid
39http://cnx.org/content/m10867/latest/sixth.mid
40http://cnx.org/content/m10867/latest/seventh.mid
41http://cnx.org/content/m10867/latest/octave.mid
42http://cnx.org/content/m10867/latest/ninth.mid
43http://cnx.org/content/m10867/latest/tenth.mid
44http://cnx.org/content/m10867/latest/eleventh.mid

139

Figure 4.29

(Solution to Exercise 4.11 on p. 168.)

4.5.3 Classifying Intervals

So far, the actual distance, in half-steps, between the two notes has not mattered. But a third made
up of three half-steps sounds di erent from a third made up of four half-steps. And a fth made
up of seven half-steps sounds very di erent from one of only six half-steps. So in the second step
of identifying an interval, clef (Section 1.1.2), key signature (Section 1.1.4), and accidentals (pg 17)
become important.

Figure 4.30: A to C natural and A to C sharp are both thirds, but A to C sharp is a larger

interval, with a di erent sound. The di erence between the intervals A to E natural and A to E
at is even more noticeable.

Listen to the di erences in the thirds45 and the fths46 in Figure 4.30.

So the second step to naming an interval is to classify it based on the number of half steps
(Section 4.2) in the interval. Familiarity with the chromatic scale (pg 123) is necessary to do this
accurately.

4.5.3.1 Perfect Intervals

perfectPrimes, octaves, fourths, and fths can be intervals.

45http://cnx.org/content/m10867/latest/twothirds.mid
46http://cnx.org/content/m10867/latest/two fths.mid

140 CHAPTER 4. NOTES AND SCALES

Note: These intervals are never classi ed as major or minor, although they can be

augmented or diminished (see below (Section 4.5.3.3)).

acousticsWhat makes these particular intervals perfect? The physics of sound waves ( ) shows us

that the notes of a perfect interval are very closely related to each other. (For more information on

this, see Frequency, Wavelength, and Pitch 47 and Harmonic Series48.) Because they are so closely

related, they sound particularly good together, a fact that has been noticed since at least the times

of classical Greece, and probably even longer. (Both the octave and the perfect fth have prominent

positions in most of the world's musical traditions.) Because they sound so closely related to each

other, they have been given the name "perfect" intervals.

Note: Actually, modern equal temperament (Section 6.2.3.2) tuning does not give the
harmonic-series-based pure (Section 6.2.2.1) perfect fourths and fths. For the music-
theory purpose of identifying intervals, this does not matter. To learn more about how
tuning a ects intervals as they are actually played, see Tuning Systems (Section 6.2).

unisonA perfect prime is also called a . It is two notes that are the same pitch (Section 1.1.3).

A perfect octave is the "same" note an octave (Section 4.1) - 12 half-steps - higher or lower. A

perfect 5th is 7 half-steps. A perfect fourth is 5 half-steps.

Example 4.6:

Perfect Intervals

Figure 4.31

Listen to the octave49, perfect fourth50, and perfect fth51.

4.5.3.2 Major and Minor Intervals
Seconds, thirds, sixths, and sevenths can be major intervals or minor intervals. The minor

interval is always a half-step smaller than the major interval.

Major and Minor Intervals

• 1 half-step = minor second (m2)
• 2 half-steps = major second (M2)
• 3 half-steps = minor third (m3)
• 4 half-steps = major third (M3)
47"Frequency, Wavelength, and Pitch" <http://cnx.org/content/m11060/latest/>
48"Harmonic Series" <http://cnx.org/content/m11118/latest/>
49http://cnx.org/content/m10867/latest/P8.mp3
50http://cnx.org/content/m10867/latest/P4.mp3
51http://cnx.org/content/m10867/latest/P5.mp3

141

• 8 half-steps = minor sixth (m6)
• 9 half-steps = major sixth (M6)
• 10 half-steps = minor seventh (m7)
• 11 half-steps = major seventh (M7)

Example 4.7:

Major and Minor Intervals

Figure 4.32

Listen to the minor second52, major second53, minor third54, major third55, minor sixth56,
major sixth57, minor seventh58, and major seventh59.

52http://cnx.org/content/m10867/latest/min2.mp3
53http://cnx.org/content/m10867/latest/M2.mp3
54http://cnx.org/content/m10867/latest/min3.mp3
55http://cnx.org/content/m10867/latest/M3.mp3
56http://cnx.org/content/m10867/latest/min6.mp3
57http://cnx.org/content/m10867/latest/M6.mp3
58http://cnx.org/content/m10867/latest/min7.mp3
59http://cnx.org/content/m10867/latest/M7.mp3

142 CHAPTER 4. NOTES AND SCALES

Exercise 4.12:

Give the complete name for each interval.

Figure 4.33

(Solution to Exercise 4.12 on p. 168.)

Exercise 4.13:

Fill in the second note of the interval given.

Figure 4.34

143

(Solution to Exercise 4.13 on p. 169.)

4.5.3.3 Augmented and Diminished Intervals

augmentedIf an interval is a half-step larger than a perfect or a major interval, it is called . An

diminishedinterval that is a half-step smaller than a perfect or a minor interval is called . A double

sharp (pg 17) or double at (pg 17) is sometimes needed to write an augmented or diminished

interval correctly. Always remember, though, that it is the actual distance in half steps between the

notes that determines the type of interval, not whether the notes are written as natural, sharp, or

double-sharp.

Example 4.8:

Some Diminished and Augmented Intervals

Figure 4.35

Listen to the augmented prime60, diminished second61, augmented third62, diminished
sixth63, augmented seventh64, diminished octave65, augmented fourth66, and diminished
fth67. Are you surprised that the augmented fourth and diminished fth sound the same?

60http://cnx.org/content/m10867/latest/aug1.mid
61http://cnx.org/content/m10867/latest/dim2.mid
62http://cnx.org/content/m10867/latest/aug3.mid
63http://cnx.org/content/m10867/latest/dim6.mid
64http://cnx.org/content/m10867/latest/aug7.mid
65http://cnx.org/content/m10867/latest/dim8.mid
66http://cnx.org/content/m10867/latest/aug4.mid
67http://cnx.org/content/m10867/latest/dim5.mid

144 CHAPTER 4. NOTES AND SCALES

Exercise 4.14:

Write a note that will give the named interval.

Figure 4.36

(Solution to Exercise 4.14 on p. 169.)

As mentioned above, the diminished fth and augmented fourth sound the same. Both are six

tritonehalf-steps, or three whole tones, so another term for this interval is a . In Western Music

(Section 2.8), this unique interval, which cannot be spelled as a major, minor, or perfect interval, is

considered unusually dissonant (Section 5.3) and unstable (tending to want to resolve (pg 185) to

another interval).

You have probably noticed by now that the tritone is not the only interval that can be "spelled"

in more than one way. In fact, because of enharmonic spellings (Section 1.1.5), the interval for any

two pitches can be written in various ways. A major third could be written as a diminished fourth,

for example, or a minor second as an augmented prime. Always classify the interval as it is written;
the composer had a reason for writing it that way. That reason sometimes has to do with subtle

di erences in the way di erent written notes will be interpreted by performers, but it is mostly a

matter of placing the notes correctly in the context of the key (Section 4.3), the chord (Chords),

and the evolving harmony (Section 2.5). (Please see Beginning Harmonic Analysis (Section 5.5) for

more on that subject.)